First, by theorem 2.1.1 in [1]}
\(\begin{pmatrix} \dfrac{U_F\left(\frac{1}{1- \exp \left( {-\frac{E_1^*}{N}}\right)} \right) -\beta _N}{\alpha _N} , & \dots , & \dfrac{U_F\left(\frac{1}{1- \exp \left({-\frac{E_1^*}{N} - \frac{E_2^*}{N-1} - \dots - \frac{E_{q+1}^*}{N-q} }\right)} \right)-\beta _N}{\alpha _N}\end{pmatrix} \\ \Rightarrow \begin{pmatrix}\dfrac{(E_1^*)^{-\gamma }-1 }{\gamma }, \dfrac{(E_1^*+E_2^*)^{-\gamma }-1}{\gamma }, \dots , \dfrac{(E_1^*+E_2^*+ \dots + E^*_{q+1})^{-\gamma } -1 }{\gamma }\end{pmatrix}.\) 
